Eleven-Dimensional Supergravity in 4D, $N=1$ Superspace

We give a formulation of linearized 11D supergravity in 4D, $N=1$ superspace keeping all eleven bosonic coordinates. The fields are fluctuations around $\mathbf{M}={\mathbf{R}}^{4\mathrm{\mid }4}\times Y$, where $Y$ is a background Riemannian 7-manifold admitting a $G_2$ structure. We embed the 11D fields into superfield representations of the 4D, $N=1$ superconformal algebra. These consist of the conformal graviton superfield, seven conformal gravitino superfields, a tensor hierarchy of superfields describing the 11D 3-form, and a non-abelian Kaluza-Klein vector multiplet gauging the tensor hierarchy by diffeomorphisms on $Y$. The quadratic action consists of the linearization of a superspace volume term and a Chern-Simons action for the gauged hierarchy coupled to the supergravity and gravitino superfields, and the full structure is fixed by superconformal and gauge invariance. When this action is projected to components, we recover the full linearized action of 11D supergravity.